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A020704 Pisot sequences E(3,10), P(3,10). 1
3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243, 16835050, 55602393, 183642229, 606529080, 2003229469, 6616217487, 21851881930, 72171863277, 238367471761, 787274278560, 2600190307441, 8587845200883, 28363725910090 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

REFERENCES

Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = 3*a(n-1) + a(n-2) (holds at least up to n = 1000 but is not known to hold in general).

Conjectures from Colin Barker, Jun 05 2016: (Start)

a(n) = (2^(-1-n)*((3-sqrt(13))^n*(-11+3*sqrt(13)) + (3+sqrt(13))^n*(11+3*sqrt(13))))/sqrt(13).

G.f.: (3+x) / (1-3*x-x^2).

(End)

Theorem: For E(3,10), a(n) = 3 a(n - 1) +  a(n - 2) for n>=2. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016

MATHEMATICA

RecurrenceTable[{a[0] == 3, a[1] == 10, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 05 2016 *)

PROG

(MAGMA) Exy:=[3, 10]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..30]]; // Bruno Berselli, Feb 05 2016

CROSSREFS

This is a subsequence of A006190.

See A008776 for definitions of Pisot sequences.

Sequence in context: A271943 A255116 A006190 * A113299 A126931 A257178

Adjacent sequences:  A020701 A020702 A020703 * A020705 A020706 A020707

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified March 28 02:00 EDT 2017. Contains 284182 sequences.